Factoring Higher-Dimensional Shifts Of Finite Type Onto The Full Shift

A one-dimensional shift of finite type with entropy at least log factors onto the full -shift. The factor map is constructed by exploiting the fact that , or a subshift of , is conjugate to a shift of finite type in which every symbol can be followed by at least symbols. We will investigate analogous statements for higher-dimensional shifts of finite type. We will also show that for a certain class of mixing higher-dimensional shifts of finite type, sufficient entropy implies that is finitely equivalent to a shift of finite type that maps onto the full -shift

Auslander Systems

The authors generalize the dynamical system constructed by J. Auslander in 1959, resulting in perhaps the simplest family of examples of minimal but not strictly ergodic systems. A characterization of unique ergodicity and mean-L-stability is given. The new systems are also shown to have zero topological entropy and fail to be weakly rigid. Some results on the set of idempotents in the enveloping semigroup are also achieved

The Symbolic Dynamics Of Multidimensional Tiling Systems

We prove a multidimensional version of the theorem that every shift of finite type has a power that can be realized as the same power of a tiling system. We also show that the set of entropies of tiling systems equals the set of entropies of shifts of finite type

IP Cluster Points Idempotents, And Recurrent Sequences

We consider the dynamical system (M,S) where M is the orbit closure of a nonperiodic recurrent sequence of O\u27s and 1\u27s (for example, the Morse sequence) and S is the shift map. The enveloping semigroup is E(M) = {Sⁿ : n ∈ Z} where the closure is taken in the topology of pointwise convergence. H. Furstenberg was the first to establish the existence of relationships between recurrence, IP sets, and idempotents in the enveloping semigroup, and the first author has proven that the closure of the set of idempotents coincides with the IP cluster points. In this paper the authors compute this set for (M,S) and shed light on other combinatorial properties of generalized Morse sequences

Putting The Pieces Together: Understanding Robinson’s Nonperiodic Tilings

A discussion of Robinson\u27s nonperiodic tilings and nonperiodic tilings with nonsquare tiles (Penrose and pinwheel)

Renewal Systems, Sharp-Eyed Snakes, And Shifts Of Finite Type

Johnson and Madden look at collections of bi-infinite strings of symbols that occur in several different areas of mathematics and ask whether these collections are the same in some sense. A dynamical systems property called entropy can be used to show that the shifts of finite type are not all conjugate to uniquely decipherable renewal systems

The Decomposition Theorem For Two-Dimensional Shifts Of Finite Type

A one-dimensional shift of finite type can be described as the collection of bi-infinite walks along an edge graph. The Decomposition Theorem states that every conjugacy between two shifts of finite type can be broken down into a finite sequence of splittings and amalgamations of their edge graphs. When dealing with two-dimensional shifts of finite type, the appropriate edge graph description is not as clear; we turn to Nasu\u27s notion of a textile system for such a description and show that all two-dimensional shifts of finite type can be so described. We then define textile splittings and amalgamations and prove that every conjugacy between two-dimensional shifts of finite type can be broken down into a finite sequence of textile splittings, textile amalgamations, and a third operation called an inversion

The Relative Growth On Information In Two-Dimensional Partitions

Let ... ∈ [0, 1)^sup 2^. In this paper we find the rate at which knowledge about the partition elements ... lies in for one sequence of partitions determines the partition elements it lies in for another sequence of partitions. This rate depends on the entropy of these partitions and the geometry of their shapes, and gives a two-dimensional version of Lochs\u27 theorem

Projectional Entropy In Higher Dimensional Shifts Of Finite Type

Any higher dimensional shift space (X, ℤᵈ) contains many lower dimensional shift spaces obtained by projection onto r-dimensional sublattices L of ℤᵈ where r \u3c d. We show here that any projectional entropy is bounded below by the ℤᵈ entropy and, in the case of certain shifts of finite type satisfying a mixing condition, equality is achieved if and only if the shift of finite type is the infinite product of a lower dimensional projection

Creating research-ready partnerships: The initial development of seven implementation laboratories to advance cancer control

BACKGROUND: In 2019-2020, with National Cancer Institute funding, seven implementation laboratory (I-Lab) partnerships between scientists and stakeholders in \u27real-world\u27 settings working to implement evidence-based interventions were developed within the Implementation Science Centers in Cancer Control (ISC3) consortium. This paper describes and compares approaches to the initial development of seven I-Labs in order to gain an understanding of the development of research partnerships representing various implementation science designs. METHODS: In April-June 2021, members of the ISC3 Implementation Laboratories workgroup interviewed research teams involved in I-Lab development in each center. This cross-sectional study used semi-structured interviews and case-study-based methods to collect and analyze data about I-Lab designs and activities. Interview notes were analyzed to identify a set of comparable domains across sites. These domains served as the framework for seven case descriptions summarizing design decisions and partnership elements across sites. RESULTS: Domains identified from interviews as comparable across sites included engagement of community and clinical I-Lab members in research activities, data sources, engagement methods, dissemination strategies, and health equity. The I-Labs use a variety of research partnership designs to support engagement including participatory research, community-engaged research, and learning health systems of embedded research. Regarding data, I-Labs in which members use common electronic health records (EHRs) leverage these both as a data source and a digital implementation strategy. I-Labs without a shared EHR among partners also leverage other sources for research or surveillance, most commonly qualitative data, surveys, and public health data systems. All seven I-Labs use advisory boards or partnership meetings to engage with members; six use stakeholder interviews and regular communications. Most (70%) tools or methods used to engage I-Lab members such as advisory groups, coalitions, or regular communications, were pre-existing. Think tanks, which two I-Labs developed, represented novel engagement approaches. To disseminate research results, all centers developed web-based products, and most (n = 6) use publications, learning collaboratives, and community forums. Important variations emerged in approaches to health equity, ranging from partnering with members serving historically marginalized populations to the development of novel methods. CONCLUSIONS: The development of the ISC3 implementation laboratories, which represented a variety of research partnership designs, offers the opportunity to advance understanding of how researchers developed and built partnerships to effectively engage stakeholders throughout the cancer control research lifecycle. In future years, we will be able to share lessons learned for the development and sustainment of implementation laboratories